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 KISS FFT - A mixed-radix Fast Fourier Transform based up on the principle, 
 "Keep It Simple, Stupid."
 
     There are many great fft libraries already around.  Kiss FFT is not trying
 to be better than any of them.  It only attempts to be a reasonably efficient, 
 moderately useful FFT that can use fixed or floating data types and can be 
 incorporated into someone's C program in a few minutes with trivial licensing.
 
 USAGE:
 
     The basic usage for 1-d complex FFT is:
 
         #include "kiss_fft.h"
 
         kiss_fft_cfg cfg = kiss_fft_alloc( nfft ,is_inverse_fft ,0,0 );
 
         while ...
         
             ... // put kth sample in cx_in[k].r and cx_in[k].i
             
             kiss_fft( cfg , cx_in , cx_out );
             
             ... // transformed. DC is in cx_out[0].r and cx_out[0].i 
             
         free(cfg);
 
     Note: frequency-domain data is stored from dc up to 2pi.
     so cx_out[0] is the dc bin of the FFT
     and cx_out[nfft/2] is the Nyquist bin (if exists)
 
     Declarations are in "kiss_fft.h", along with a brief description of the 
 functions you'll need to use. 
 
 Code definitions for 1d complex FFTs are in kiss_fft.c.
 
 You can do other cool stuff with the extras you'll find in tools/
 
     * multi-dimensional FFTs 
     * real-optimized FFTs  (returns the positive half-spectrum: (nfft/2+1) complex frequency bins)
     * fast convolution FIR filtering (not available for fixed point)
     * spectrum image creation
 
 The core fft and most tools/ code can be compiled to use float, double 
 or Q15 short samples. The default is float.
 
 
 BACKGROUND:
 
     I started coding this because I couldn't find a fixed point FFT that didn't 
 use assembly code.  I started with floating point numbers so I could get the 
 theory straight before working on fixed point issues.  In the end, I had a 
 little bit of code that could be recompiled easily to do ffts with short, float
 or double (other types should be easy too).  
 
     Once I got my FFT working, I was curious about the speed compared to
 a well respected and highly optimized fft library.  I don't want to criticize 
 this great library, so let's call it FFT_BRANDX.
 During this process, I learned:
 
     1. FFT_BRANDX has more than 100K lines of code. The core of kiss_fft is about 500 lines (cpx 1-d).
     2. It took me an embarrassingly long time to get FFT_BRANDX working.
     3. A simple program using FFT_BRANDX is 522KB. A similar program using kiss_fft is 18KB (without optimizing for size).
     4. FFT_BRANDX is roughly twice as fast as KISS FFT in default mode.
 
     It is wonderful that free, highly optimized libraries like FFT_BRANDX exist.
 But such libraries carry a huge burden of complexity necessary to extract every 
 last bit of performance.
 
     Sometimes simpler is better, even if it's not better.
 
 FREQUENTLY ASKED QUESTIONS:
         Q: Can I use kissfft in a project with a ___ license?
         A: Yes.  See LICENSE below.
 
         Q: Why don't I get the output I expect?
         A: The two most common causes of this are 
                 1) scaling : is there a constant multiplier between what you got and what you want?
                 2) mixed build environment -- all code must be compiled with same preprocessor 
                 definitions for FIXED_POINT and kiss_fft_scalar
 
         Q: Will you write/debug my code for me?
         A: Probably not unless you pay me.  I am happy to answer pointed and topical questions, but 
         I may refer you to a book, a forum, or some other resource.
 
 
 PERFORMANCE:
     (on Athlon XP 2100+, with gcc 2.96, float data type)
 
     Kiss performed 10000 1024-pt cpx ffts in .63 s of cpu time.
     For comparison, it took md5sum twice as long to process the same amount of data.
 
     Transforming 5 minutes of CD quality audio takes less than a second (nfft=1024). 
 
 DO NOT:
     ... use Kiss if you need the Fastest Fourier Transform in the World
     ... ask me to add features that will bloat the code
 
 UNDER THE HOOD:
 
     Kiss FFT uses a time decimation, mixed-radix, out-of-place FFT. If you give it an input buffer  
     and output buffer that are the same, a temporary buffer will be created to hold the data.
 
     No static data is used.  The core routines of kiss_fft are thread-safe (but not all of the tools directory).
 
     No scaling is done for the floating point version (for speed).  
     Scaling is done both ways for the fixed-point version (for overflow prevention).
 
     Optimized butterflies are used for factors 2,3,4, and 5. 
 
     The real (i.e. not complex) optimization code only works for even length ffts.  It does two half-length
     FFTs in parallel (packed into real&imag), and then combines them via twiddling.  The result is 
     nfft/2+1 complex frequency bins from DC to Nyquist.  If you don't know what this means, search the web.
 
     The fast convolution filtering uses the overlap-scrap method, slightly 
     modified to put the scrap at the tail.
 
 LICENSE:
     Revised BSD License, see COPYING for verbiage. 
     Basically, "free to use&change, give credit where due, no guarantees"
     Note this license is compatible with GPL at one end of the spectrum and closed, commercial software at 
     the other end.  See http://www.fsf.org/licensing/licenses
 
     A commercial license is available which removes the requirement for attribution.  Contact me for details.
 
   
 TODO:
     *) Add real optimization for odd length FFTs 
     *) Document/revisit the input/output fft scaling
     *) Make doc describing the overlap (tail) scrap fast convolution filtering in kiss_fastfir.c
     *) Test all the ./tools/ code with fixed point (kiss_fastfir.c doesn't work, maybe others)
 
 AUTHOR:
     Mark Borgerding
     Mark@Borgerding.net